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Abstract

We report on a study of electromagnetic waves propagation in thin periodically ordered photonic nanostructures in the spectral range where the light wavelength is on the order of the lattice parameter. The vector KKR method we use allows us to determine the group index from finite photonic structures including extinction providing confirmation of recently emerged results. We show that for certain frequencies the group velocity of opal slabs can either be superluminal or approach zero depending on the crystal thickness and the unavoidable presence of losses. In some cases, group velocity can be negative. Such behavior can be clearly attributed to the finite character of the three-dimensional structure and reproduces previously reported experimental observations. Calculations show that contrary to the predictions of extraordinary group velocity reductions for infinite periodic structures, the group velocity of real opals may exhibit strong fluctuations at the high energy range. Hence, a direct identification between the calculated anomalous group velocities, for an actual opal film, and the predicted propagating low dispersion modes for an ideal infinite ordered structure seems difficult to establish.

Figures (6)

High-energy band structure in the ΓL direction of an ideal close-packed fcc opal made of polystyrene spheres (εs=2.5) in air (ε=1). Energy is expressed in reduced frequency units, lattice parameter, a, over wavelength in vacuum, λ. The shaded region corresponds to the considered high energy range of the spectrum, 1.05 ≤ a/λ ≤1.35.

a) Reflected modes for a glass supported opal made of 10 [111] layers, taking εsph=2.5+0.1i for the dielectric spheres and εg=2.34 for the glass substrate. b) Transmitted modes for the same structure. The vertical dotted lines corresponds to the diffraction cut-off for the six lower order diffracted modes and the next six respectively. All curves are plotted as a function of frequency in reduced, a/λ, units.

a) Phase delay, in units of π, introduced to a forwardly transmitted beam propagating along the [111] direction of a 10 [111]-layers opal made of spheres, for increasing values of the dielectric contrast. Τhe imaginary part of the sphere-dielectric function is kept as constant. b) Corresponding group indexes. All curves are plotted as a function of frequency in reduced, a/λ, units.

Phase delay divided by the number of [111] planes, in units of π, introduced to a forwardly transmitted field propagating in the ΓL of the fcc glass supported (εg=2.34) structure. All phases are plotted as a function of frequency in reduced, a/λ, units. Nanospheres dielectric function εsph=2.5+0.04i. The number of [111] planes is indicated in the inset of the figure.

Group index for different opal thicknesses; all plotted as a function of frequency in reduced units. Nanospheres dielectric function εsph=2.5+0.04i. The number of [111] planes is indicated in the inset of the figure.

Calculated group index for different opal thickness as a function of reduced frequency in the A region. The graphic inset corresponds to the number of [111] layers. The dielectric constant of the spheres and the glass support are εshp=2.6+0.0575i and εg=2.34, respectively [21].